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In the case of negligible quantum effects, the microscopic behavior of the system can be described within the Boltzmann transport equation. Otherwise, equations based on the Wigner-Boltzmann formula must be considered [153]. The BTE allows for the description of carriers motion in an inhomogeneous material with an arbitrary band structure to external and internal forces. The main components for mathematical models for semiconductor devices are derived from this semi-classical fundamental equation which reads [154]
The solution of Boltzmann's equation is the time dependent carrier distribution function f(r,k,t) in the six-dimensional phase space (three dimensions in real space r and three dimensions in momentum space k. The right hand side of the BTE contains the rate of change of the distribution function due to collisions. The electron velocity vis determined from the band structure for the semiconductor under consideration. The electric field F(r,t) can be determined via Poisson's equation [155]
where Ndd is the differential doping density and ε is the permittivity of the material. The electron and hole concentrations, n and p, are the lowest order moments of the distribution function. If the distribution function is a known solution of (2.1), simultaneously with the Poisson equation, macroscopic quantities of interest such as the electron density, drift velocity and mean energy can be calculated according to
Due to its complicated nature, a general analytical solution of BTE can not be found. On the other hand, attempts to solve this equation numerically by discretization of the differential and integral operators is computationally very expensive. A widely used numerical method for solving the Boltzmann equation is the Monte Carlo approach. The MC method has been proven to produce reliable results but is computationally expensive. Moreover, if the distribution of high-energetic carriers is relevant, or if the carrier concentration is very low in specific regions of the device, Monte Carlo simulations produce a high variance in the results. Therefore, a common simplification is to investigate only some moments of the distribution function, such as the carrier concentration and the carrier temperature. In other words, the Boltzmann equation is simplified to the models, where the mentioned physical quantities are considered as the ensemble averages at the equilibrium state. For example, the carrier temperatures can be assumed to be in equilibrium with the lattice temperature. These approaches are usually limited in their predictive capability because they employ a number of fitting parameters used to match experimental data. Below, a short overview of the most widely used transport models is presented.
